Answer:
The rabbit should be fed:
grams of food A
grams of food B
grams of food C
For
.
Explanation:
This can be solved by a system of equations.
I am going to say that x is the number of grams of food A, y is the number of grams of food B and z is the number of grams of Food C.
The problem states that:
A researcher wants to provide a rabbit exactly 162 units of protein:
There are 5 units of protein in each gram of food A, 11 units of protein in each gram of food B and 23 units of protenin in each gram of food C. So

A researcher wants to provide a rabbit exactly 72 units of carbohydrates:
There are 2 units of carbohydrates in each gram of food A, 5 units of carbohydrates in each gram of food B and 11 units of carbohydrates in each gram of food C. So:

A researcher wants to provide a rabbit exactly 30 units of Vitamin A:
There is 1 unit of Vitamin A in each gram of food A, 2 units of Vitamin A in each gram of food B and 4 units of Vitamin A in each gram of food C. So:
.
We have to solve the following system of equations:


.
I think that the easier way to solve this is reducing the augmented matrix of this system.
This system has the following augmented matrix:
![\left[\begin{array}{cccc}5&11&23&162\\2&5&11&72\\1&2&4&30\end{array}\right]](https://img.qammunity.org/2020/formulas/mathematics/college/m0kdkir0gct8s9kogimc4iyglzgsl94abp.png)
To help reduce this matrix, i am going to swap the first line with the third

Now we have the following matrix:
![\left[\begin{array}{cccc}1&2&4&30\\2&5&11&72\\5&11&23&162\end{array}\right]](https://img.qammunity.org/2020/formulas/mathematics/college/n6vqgi5461xes24a1wywph9avu9y6ieqko.png)
Now i am going to do these following operations, to reduce the first row:


Now we have
![\left[\begin{array}{cccc}1&2&4&30\\0&1&3&12\\0&1&3&12\end{array}\right]](https://img.qammunity.org/2020/formulas/mathematics/college/7s6e7sy6k8oebyod79p4xkvjkitx7ea92e.png)
Now, to reduce the second row, i do:

The matrix is:
![\left[\begin{array}{cccc}1&2&4&30\\0&1&3&12\\0&0&0&0\end{array}\right]](https://img.qammunity.org/2020/formulas/mathematics/college/t581a1jppmmerex6foqkq6ls54ngw182cm.png)
This means that z is a free variable, so we are going to write y and x as functions of z.
From the second line, we have


From the first line, we have




Our solution is:
.
However, we can not give a negative number of grams of a food. So





The rabbit should be fed:
grams of food A
grams of food B
grams of food C
For
.